Only in the games it seems because that's not the case in the comics where he has his own adventures as king.

I wonder why, it's not the same in the games.
That puzzles the hell out of me as well. I guess that's why I expected more screen time. Plus, we were spending a lot of time in Ferelden. So yeah, that's pretty weird. At least, regardless of world state, Alistair had at least a handful of lines in DA2.
For King Alistair in DA2, aside from eye-rolling comments like "Swooping is bad." <winks to the camera> and "Old Ball and Chain," he does add a little bit to the story. He mentions that Orlais is "having troubles of it's own." Which isn't a lot, but it's at least giving the player an idea of what could be happening outside of Kirkwall.
King Alistair in this game doesn't help further the story much other than kicking the mages out of Ferelden. Really, other than his more pro-Mage leanings, it could have been just about any generic ruler kicking Fiona and her crew out.
Drunk Alistair probably doesn't show up because Bann Teagan takes him home to help him sober up in Act 3 of DA2. It still would have been nice though.